Second Derivatives of Implicit Functions

Elliptical structures, like arches in bridges, are often represented by implicit equations, since height cannot be easily expressed as a function of the horizontal dimension.

In these situations, implicit differentiation helps determine how the slope changes along the arch.

Since y depends on x, the chain rule is applied to differentiate the ellipse’s equation. This gives the first derivative, showing that the slope depends on both coordinates.

To find the second derivative, differentiate the first derivative again using the quotient rule, as the numerator and the denominator depend on x.

This gives a new expression that still contains the first derivative itself. Next, substitute the full expression for the first derivative back into the unsimplified equation.

Simplify the complex numerator by finding a common denominator to combine terms.

Rearranging the original ellipse equation provides an identity for that combined term in the numerator. Substituting this identity yields the final expression for the second derivative.

A positive second derivative shows concave-up curvature, while a negative one gives concave-down curvature—showing how slope and curvature change, which help assess the stability of arches.